✏️ Payback Period when you must prorate

Payback period questions like the following are fairly straightforward:

❔ Consider the following cash flows. What is your payback period?

T	CF
0	-10
1	4
2	3
3	3
4	2

✔ Click here to view answer

I like to add a “running total” column:

T	CF	Running total
0	-10	-10
1	4	-6
2	3	-3
3	3	0
4	2	2

You payback period in this case is exactly 3 years.

But what about a question in which you don’t get paid back exactly on a year boundary?

❔ Consider the following cash flows. What is your payback period?

T	CF
0	-10
1	4
2	3
3	5
4	2

✔ Click here to view answer

I suggest 3 steps:

1. Find the year when you break even Viewing the data below, our running total is negative at $T=2$ and positive at $T=3$, so we break even at some point during the third year.
2. How what are the cash flows in that year? What cash flows are required to break even? Cash flows in year 3: $5M
   Cash flows to break even: $3M
3. Divide. It takes $\frac{3}{5} = 60\%$ of the third year to pay back.
4. Finalize the answer Payback period is $2+60\% = 2.6 \;years.$

T	CF	Running total
0	-10	-10
1	4	-6
2	3	-3
3	5	2
4	2	4

This does rely on the simplifying assumption that you receive the cash flows evenly throughout the year, so if you are told that you get the $5M as a lump sum at $T=3$, then your payback period will be three. Typically, however, you will be receiving the money throughout the year, and an “even cash flows” assumption is appropriate. Bruce will work hard to make it clear, and I’m happy to clarify.